Lesson Objectives
• Learn how to find the measure of unknown angles using geometric properties
• Learn how to find the measure of unknown angles using the angle sum property of triangles
• Learn how to find missing angle measures and side lengths in similar triangles

## Angle Relationships

In this lesson, we will explore a few basic geometric properties, learn how to find the measure of certain angles, and learn how to find the side lengths in similar triangles.

### Vertical Angles

Vertical angles are angles that are opposite of each other where two lines cross. Vertical angles have equal measures. Let's look at an example.
Example #1: Find the measure of each marked angle. The two angles given are vertical angles. These angles are opposite of each other where two lines cross. By definition, the two angles will have the same measure, therefore, we can set the two given expressions equal to each other and solve. $$10x + 35=20x - 55$$ $$-10x=-90$$ $$x=9$$ Now we need to substitute into either to find our angle measurements. $$10(9) + 35=90 + 35=125$$ The two angles measure 125°

### Transversal and Parallel Lines

Parallel lines are lines that lie in the same plane and will never intersect. A transversal is any line that intersects two other lines in the same plane. When the transversal intersects two parallel lines, eight angles are formed with certain properties.
Name Rule
Alternate Interior AnglesAngle Measures are Equal
Alternate Exterior AnglesAngle Measures are Equal
Interior Angles on Same Side of TransversalAngle Measures Add to 180°
Corresponding AnglesAngle Measures are Equal
Interior angles are the angles formed between the parallel lines, whereas, the exterior angles are formed outside of the parallel lines. Let's look at an example.
Example #2: Find the measure of all 8 angles.
Lines m and n are parallel.
Angle 2 has a measure of (11x - 53)°
Angle 8 has a measure of (4x + 8)° Angles 2 and 7 are alternate exterior angles and have the same measure. Angles 7 and 8 form a straight angle, which has a measure of 180°. Since we know the measures of angles 2 and 7 are the same and the measures of angles 7 and 8 sum to 180°, we set the sum of our two expressions given to 180° and solve. $$11x - 53 + 4x + 8=180$$ $$15x - 45=180$$ $$15x=225$$ $$x=15$$ Now, we can plug back into our expressions. $$11(15) - 53=112$$ This tells us that angle 2 has a measure of 112°. Since angles 2 and 3 are vertical angles, they will have the same measure. Additionally, angles 2 and 7 are alternate exterior angles and have the same measure. Lastly, angles 6 and 7 are vertical angles and have the same measure. We can state that the measures of angles 2, 3, 6, and 7 is 112°. Additionally, we know that our other four angles will be found as 180° - 112° or 68°. So the measures of angles 1, 4, 5, and 8 is 68°.

### Angle Sum Property of a Triangle

The sum of the measures of the angles of any triangle is 180°. We can use this property to solve for unknown angle measures. Let's look at an example.
Example #3: Find the measure of all angles.
Angle 1 has a measure of (6x + 1)°
Angle 2 has a measure of (10x - 56)°
Angle 3 has a measure of 43° Since the sum of the angles in a triangle is 180°, we can set the sum of the three angle measures equal to 180 and solve. $$6x + 1 + 10x - 56 + 43=180$$ $$16x - 12=180$$ $$16x=192$$ $$x=12$$ Angle 1: $$(6(12) + 1)°=73°$$ Angle 2: $$(10(12) - 56)°=64°$$ We can state that Angle 1 has a measure of 73°, Angle 2 has a measure of 64°, and Angle 3 has a measure of 43°.

### Similar Triangles

Similar triangles are triangles where the corresponding angles have the same measure and the corresponding sides are proportional. Let's look at an example.
Example #4: Find all unknown angle measures.
Triangles ABC and DEF are similar. Corresponding angles are marked using matching arcs. This notes that their angle measures are the same.
Angle A has a measure of 18°
Angle C has a measure of 144° Angles A and D are corresponding angles, therefore, angle D has a measure of 18°. Angles C and F are corresponding angles, therefore, angle F has a measure of 144°. For angles B and E, they are corresponding angles and can be found by subtracting 180 - 144 - 18. 180 - 162 is 18, therefore, the measure of angles B and E is 18°.
In other cases, we will be asked to find the missing length. Let's look at an example.
Example #5: Find the missing side lengths in triangle DEF.
Triangles ABC and DEF are similar.
Side AB has length 15.
Side BC has length 30.5.
Side AC has length 24.
Side DE has length 30. Since we know that side DE of triangle DEF corresponds to side AB of triangle ABC, and sides EF and BC correspond, we can set up the following proportion: $$\frac{DE}{AB}=\frac{EF}{BC}$$ Plug in what is given: $$\frac{30}{15}=\frac{EF}{30.5}$$ Cross Multiply: $$15EF=915$$ Divide both sides by 15: $$EF=61$$ Since we know that side DF of triangle DEF corresponds to side AC of triangle ABC, and sides EF and BC correspond, we can set up the following proportion: $$\frac{\overline{DE}}{\overline{AB}}=\frac{\overline{DF}}{\overline{AC}}$$ Plug in what is given: $$\frac{30}{15}=\frac{\overline{DF}}{24}$$ Cross Multiply: $$15\overline{DF}=720$$ Divide both sides by 15: $$\overline{DF}=48$$ Side DF has length 48.

#### Skills Check:

Example #1

Find the measure of the marked angles. A
129°, 51°
B
60°, 120°
C
40°, 140°
D
70°, 130°
E
63°, 117°

Example #2

Find the unknown angle measures. Angle 1: (5x + 37)°
Angle 2: (2x - 15)°
Angle 3: 18°

A
100°, 25°, 55°
B
137°, 25°, 18°
C
127°, 35°, 18°
D
104°, 39°, 37°
E
155°, 12.5°, 12.5°

Example #3

Find the unknown angle measures. Triangles ABC and DEF are similar.
Angle A has a measure of 77°
Angle E has a measure of (15x + 5)°
Angle C has a measure of (14x + 11)°

A
A, E: 60°, B,F: 60°, C,D: 60°
B
A, E: 55°, B,F: 50°, C,D: 75°
C
A, D: 55°, E,B: 50°, C,F: 75°
D
A, E: 77°, B,F: 50°, C,D: 53°
E
A, D: 77°, B,E: 50°, C,F: 53°         